1Algebraic Characterisation of the H ∞ and H2 Norms for Linear Continuous-Time Periodic Systems∗

It is well-known that linear, periodically time-varying, continuous-time systems are formally equivalent to so-called lifted representations that are shift-invariant, but have spatially infinite-dimensional inputs and outputs. By shift invari-ance, corresponding frequency-domain represen-tations can be constructed. Indeed, it makes sense to use the H ∞ norm of the associated frequency-domain symbol as a measure of system size. In fact, this is equal to the L2[0,∞) induced norm of the system. As an alternative measure of system size a generalisation of the H2 norm, which char-acterises the impulsive response of the system, can also be defined. The purpose of this paper is to establish finite-dimensional, algebraic characteri-sations of these norms, for linear continuous-time periodic systems.